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Applied Mathematical Sciences Volume 93 Editors S.S.Antman DepartmentofMathematics and InstituteforPhysicalScienceandTechnology UniversityofMaryland CollegePark,MD20742-4015 USA [email protected] P.Holmes DepartmentofMechanicalandAerospaceEngineering PrincetonUniversity 215FineHall Princeton,NJ08544 [email protected] K.Sreenivasan DepartmentofPhysics andtheCourantInstituteofMathematicalSciences NewYorkUniversity 70WashingtonSquareSouth NewYorkCity,NY10012 [email protected] Advisors Greengard,L. Keener,J. Keller,J. Matkowsky,B. Pego,R. Peskin,C. Stevens,A. Stuart,A. Forfurthervolumes: http://www.springer.com/series/34 David Colton•Rainer Kress Inverse Acoustic and Electromagnetic Scattering Theory Third Edition 123 DavidColton RainerKress DepartmentMathematicalScience Institutfu¨rNumerischeundAngewandte UniversityofDelaware Mathematik Newark,DE Georg-August-Universita¨tGo¨ttingen USA Go¨ttingen,Germany ISBN978-1-4614-4941-6 ISBN978-1-4614-4942-3(eBook) DOI10.1007/978-1-4614-4942-3 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012943988 Mathematics Subject Classification (2010): 35J05, 35P25, 35R25, 35R30, 45A05, 65M30, 65R20, 65R30,76Q05,78A45 ©SpringerScience+BusinessMediaNewYork1992,1998,2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect tothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) FromRainerforMarcus As farasmyeyes can see There areshadowsapproachingme And tothoseIleft behind Iwanted you toknow You’ve alwaysshared mydeepestthoughts I’ll missyou when Igo And ohwhen I’m oldandwise Bitterwords mean littleto me Like autumnwinds willblowrightthroughme And somedayin themistof time When theyasked meifIknewyou I’d smileandsayyou were afriendof mine And thesadnesswould beliftedfrommyeyes Alan Parsons Preface to the Third Edition Since the second edition of our bookappearedfourteenyearsago, the field of inverse scattering theory has continued to be an active and growing area of applied mathematics. In this third edition we have tried to bring our book up to date by includingmanyof the new developmentsin the field that havetaken place during thisperiod.We againhavemadeno effortto coverall ofthe manynewdirections in inverse scattering theory but rather have restricted ourselves to a selection of thosedevelopmentsthatwehaveeitherparticipatedinorareanaturaldevelopment of material discussed in previous editions. We have also continued to emphasize simplicity over generality, e.g. smooth domains instead of domains with corners, isotropicmediaratherthananisotropicmedia,standardboundaryconditionsrather thanmoregeneralizedones,etc.Bysodoing,wehopethatourbookwillcontinue toserveasabasicintroductiontothefieldofinversescatteringtheory. Inordertobringourbookuptodate,considerablechangeshavebeenmadetothe secondedition.Inparticular,newsectionshavebeenaddedonthelinearsampling andfactorizationmethodsforsolvingtheinversescatteringproblemaswellasex- pandedtreatmentsofiterationmethodsanduniquenesstheoremsfortheinverseob- stacleproblem.Theseadditionshavealsorequiredustoexpandourpresentationon bothtransmission eigenvaluesandboundaryintegralequationsin Sobolevspaces. Thesechangesinturnsuggestamoreintegratedviewofinversescatteringtheory.In particular,whatwaspreviouslyreferredto as the Colton–Monkand Kirsch–Kress methods,respectively,arenowviewedastwoexamplesofwhatarecalleddecompo- sitionmethods.Fromthispointofviewthetechniquesofiteration,decomposition andsamplingformanaturaltrilogyofmethodsforsolvinginversescatteringprob- lems.Althoughafewresultsfromthesecondeditionhavebeenremovedduetothe factthatwenowconsiderthemtobeobsolete,forhistoricalreasonswehavetended todososparingly. Wehopethatthisneweditionofourbookwillcontinuetoservereaderswhoare already in the field of inverse scattering theory as well as to attract newcomersto thisbeautifulareaofappliedmathematics. Newark,Delaware DavidColton Go¨ttingen,Germany RainerKress vii Preface to the Second Edition Inthefiveyearssincethefirsteditionofthisbookappeared,thefieldofinversescat- teringtheoryhascontinuedtogrowandflourish.Hence,whentheopportunityfora secondeditionpresenteditself,wewerepleasedtohavethepossibilityofupdating ourmonographtotakeintoaccountrecentdevelopmentsinthearea.Asinthefirst edition,wehavebeenmotivatedbyourownviewofinversescatteringandhavenot attemptedto includeall ofthe manynewdirectionsin the field.However,we feel thatthisneweditionrepresentsastateoftheartoverviewofthebasicelementsof themathematicaltheoryofacousticandelectromagneticinversescattering. In addition to making minor corrections and additional comments in the text andupdatingthereferences,wehaveaddednewsectionsonNewton’smethodfor solving the inverse obstacle problem (Section 5.3), the spectral theory of the far fieldoperator(Section8.4),aproofoftheuniquenessofthesolutiontotheinverse mediumproblemfor acoustic waves (Section 10.2)and a methodfor determining the support of an inhomogeneousmedium from far field data by solving a linear integralequationofthefirstkind(Section10.7). We hope that this second edition will attract new readers to the beautiful and intriguingfieldofinversescattering. Newark,Delaware DavidColton Go¨ttingen,Germany RainerKress ix